# Matrix Types Cheat Sheet

In the field of linear algebra there are variety of different matrix types. Each has its own definition and relevance. I had trouble finding a good overview online and thought I’d compile a list myself: **This article lists a selection of matrix types as well as their definition**, mostly based on the corresponding Wikipedia articles.

A matrix is **diagonal** (wiki) if all entries outside the main diagonal are zero:

$$A_{i,j}=0\Leftarrow i\ne j$$

A **square **matrix (wiki) is a matrix with the same number of rows and columns, e.g. $n\times n$.

An **identity **matrix $\boldsymbol{I}_n$ (wiki) is a *diagonal* *square* matrix whose entries on the main diagonal are one:

$$I_{i,j}=\begin{cases}1&\text{if }i=j\\0&\text{otherwise}\end{cases}$$

A **zero **matrix $\boldsymbol{0}_{n,m}$ (wiki) is a $n\times m$ matrix whose entries are zero (respectively, a **one** matrix $\boldsymbol{1}_{n,m}$ has only one entries):

$$0_{i,j}=0$$

A **normal **matrix (wiki; always *unitary*, *Hermitian*, and *skew-Hermitian*) commutes with its conjugate transpose:

$$\boldsymbol{A}^*\boldsymbol{A}=\boldsymbol{A}\boldsymbol{A}^*$$

An **upper ****triangular** matrix (wiki) has only zero entries below its main diagonal:

$$A_{i,j}=0\Leftarrow i\lt j$$

A **lower triangular** matrix (wiki) has only zero entries above its main diagonal:

$$A_{i,j}=0\Leftarrow i\gt j$$

A **symmetric** matrix (wiki) is equal to its transpose:

$$\boldsymbol{A}=\boldsymbol{A}^\mathsf{T}$$

A **skew-symmetric **matrix (wiki) is equal to the negative of its transpose:

$$\boldsymbol{A}=-\boldsymbol{A}^\mathsf{T}$$

A **Hermitian** (or **self-adjoint**) matrix (wiki) is a complex square matrix that is equal to its own conjugate transpose:

$$\boldsymbol{H}=\boldsymbol{H}^*$$

A **skew-Hermitian** (or **antihermitian**) matrix (wiki) is a complex square matrix whose conjugate transpose is the negative of the original matrix:

$$\boldsymbol{H}=-\boldsymbol{H}^*$$

For an **invertible** (also **nonsingular** or **nondegenerate**) *square* matrix $\boldsymbol{A}$ (wiki) there exists a matrix $\boldsymbol{B}$ which is inverse to $\boldsymbol{A}$:

$$\boldsymbol{A}\boldsymbol{B}=\boldsymbol{B}\boldsymbol{A}=\boldsymbol{I}_n$$

A **singular** (or **degenerate**) matrix (wiki) is not *invertible*.

A **cofactor** matrix $\boldsymbol{C}$ (wiki; also **matrix of cofactors** or **comatrix**) of a *square* matrix $\boldsymbol{A}$ is defined such that the inverse of $\boldsymbol{A}$ is the transpose of the cofactor matrix times the reciprocal of the determinant of $\boldsymbol{A}$:

$$\boldsymbol{A}^{-1} = \frac{1}{\operatorname{det}(\boldsymbol{A})} \boldsymbol{C}^\mathsf{T}$$

The transpose of an **orthogonal** matrix (wiki) is equal to its inverse:

$$\boldsymbol{A}^\mathsf{T}=\boldsymbol{A}^{-1}\iff\boldsymbol{A}^\mathsf{T}\boldsymbol{A}=\boldsymbol{A}\boldsymbol{A}^\mathsf{T}=\boldsymbol{I}$$

A matrix is **unitary** (wiki) if its conjugate transpose $\boldsymbol{U}^*$ is also its inverse: $$\boldsymbol{U}^*\boldsymbol{U}=\boldsymbol{U}\boldsymbol{U}^*=\boldsymbol{I}$$

A *symmetric **square *real matrix $\boldsymbol{A}$ is** positive-definite** (wiki), if for every non-zero column vector $\boldsymbol{z}$, $$\boldsymbol{z}^\intercal\boldsymbol{A}\boldsymbol{z}\gt0$$ holds. For **negative-definite** matrices, $\boldsymbol{z}^\intercal\boldsymbol{A}\boldsymbol{z}\lt0$. In the complex case, the *Hermitian* matrix $\boldsymbol{H}$ satisfies $\boldsymbol{z}^*\boldsymbol{H}\boldsymbol{z}\gt0$ (or $\boldsymbol{z}^*\boldsymbol{H}\boldsymbol{z}\lt0$ respectively).

A **positive-semidefinite** (wiki; or **negative-semidefinite**) matrix is defined similarly to *positive-definite* and *negative-definite* matrices, with the difference that the greater than and less than comparisons are relaxed to allow for zero scalars as well.

An **idempotent** matrix (wiki) is a square matrix which, when multiplied by itself, yields itself:

$$\boldsymbol{A}\boldsymbol{A}=\boldsymbol{A}$$

A *square* matrix $\boldsymbol{A}$ is **diagonalizable** (or **nondefective**; wiki) if it there exists a matrix $\boldsymbol{P}$ and its inverse $\boldsymbol{P}^{-1}$ such that $$\boldsymbol{P}^{-1}\boldsymbol{A}\boldsymbol{P}$$is a *diagonal* matrix.

A **permutation matrix** (wiki) is a *square* binary matrix that has exactly one entry of one in each row and each column and zeros elsewhere. It is *orthogonal*.

A **submatrix** (wiki) of another matrix is obtained by deleting any collection of rows and/or columns from it.

A **Frobenius** matrix (wiki) is a *square* matrix with the properties (1) all entries on the main diagonal are one, (2) the entries below the main diagonal of at most one column $j’$ are arbitrary, and (3) every other entry is zero:

$$A_{i,j}=\begin{cases}

1&\text{if }i=j\\

A_{i,j}&\text{if }i<j\land j=j’\\

0&\text{otherwise}

\end{cases}$$

Post title photo by Tadas Sar on Unsplash. With some imagination, there are lots of square matrices composing the building’s wall.